NMR characterization of cylinder radii distribution using a SHORE-based regularization method.

Participants : Gonzalo Sanguinetti, Matt Hall [Centre for Medical Image Computing, Dept. Computer Science, UCL] , Daniel Alexander [Centre for Medical Image Computing, Dept. Computer Science, UCL] , Rachid Deriche.

In this work, we are interested in retrieving information about the axon diameter distributions in white matter fiber bundles using NMR, which are commonly modelled as ensembles of parallel cylinders. We add regularization to the 1D-SHORE basis which results in more stable characterization of diameter distributions. To validate the method, we simulate NMR signals using the open source toolkit CAMINO. The results illustrate the enhanced estimation accuracy given by the regularization and provide an alternative validation of the SHORE based method.

Mapping Average axon diameters under long diffusion time

Participants : Gonzalo Sanguinetti, Rachid Deriche.

This work proposes an original method to recover axon diameter distribution (ADD) parameters using nuclear magnetic resonance. White matter (WM) is modelled as a bi-compartmental medium composed of an intra axonal space where the diffusion is restricted and an extra axonal space where diffusion is hindered. Under the assumption of long diffusion time, we provide a novel and efficient model for the component of the signal due to the restricted part. This technique might be interpreted as an interesting simplification of the AxCaliber framework, which leads to a simpler model and an extremely faster acquisition protocol. To test and validate our method, we use the open-source toolkit Camino for computing Monte-Carlo simulations of NMR data and model the WM as 3D cubic environments, formed by parallel cylinders with gamma distributed radii. Promising experimental results illustrate the potential of the proposed method.

Magnitude and complex based diffusion signal reconstruction

Participants : Marco Pizzolato, Timothe Boutelier [Olea Medical, La Ciotat] , Rachid Deriche.

In Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) the modeling of the magnitude signal is complicated by the Rician distribution of the noise. It is well known that when dealing instead with the complex valued signal, the real and imaginary parts are affected by Gaussian distributed noise and their modeling can thus benefit from any estimation technique suitable for this noise distribution. We present a quantitative analysis of the difference between the modeling of the magnitude diffusion signal and the modeling in the complex domain. The noisy complex and magnitude diffusion signals are obtained for a physically realistic scenario in a region close to a restricting boundary. These signals are then fitted with the Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) bases and the reconstruction performances are quantitatively compared. The noisy magnitude signal is also fitted by taking into account the Rician distribution of the noise via the integration of a Maximum Likelihood Estimator (MLE) in the SHORE. We discuss the performance of the reconstructions as function of the Signal to Noise Ratio (SNR) and the sampling resolution of the diffusion signal. We show that fitting in the complex domain generally allows for quantitatively better signal reconstruction, also with a poor SNR, provided that the sampling resolution of the signal is adequate. This applies also when the reconstruction is compared to the one performed on the magnitude via the MLE.

Extracting a biomarker for the mean cross-sectional area from the ODF

Participants : Rutger H.j Fick, Gonzalo Sanguinetti, Rachid Deriche.

Finding new biomarkers related to the microstructure of white matter (WM) is an active area of research in the MRI community. As opposed to the usual MRI markers such as fractional anisotropy (FA), these biomarkers provide a closer insight on the tissue structure. We introduce a new microstructure based biomarker that is related to the axon diameter distribution (ADD) and can be obtained with a q-space imaging technique like DSI or MAP . This feature is related with the nature and purpose of WM paths in bothnormal and pathological conditions and is obtained from the Orientation Distribution Function (ODF) as twice its maximum value. We show that this value is related with the mean cross-sectional area (MCSA) of an ensemble of parallel axons. The same geometric feature was proposed as a scalar index of microstructure, but was not related to the ODF. In this work we give the formal relation between this microstructure feature and the ODF, and validate it using state-of-the-art numerical simulations.

An Analytical 3D Laplacian Regularized SHORE Basis and Its Impact on EAP reconstruction and Microstructure Recovery

Participants : Rutger H.j Fick, Demian Wassermann, Gonzalo Sanguinetti, Rachid Deriche.

In diffusion MRI, the reconstructed Ensemble Average Propagator (EAP) from the diffusion signal provides detailed insights on the diffusion process and the underlying tissue microstructure. Recently, the Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) basis was proposed as a promising method to reconstruct the EAP. However, the fitting of the basis is sensitive to noise. To solve this we propose to use the Laplacian of the SHORE basis as a natural regularization functional. We provide the derivation of the Laplacian functional and compare its effect on EAP reconstruction with that of separated regularization of the radial and angular parts of the SHORE basis. To find optimal regularization weighting we use generalized cross-validation and validate our method quantitatively on synthetic and qualitatively on human data from the Human Connectome Project. We show that Laplacian regularization provides more accurate estimation of the signal and EAP based microstructural measures.

Using 3D-SHORE and MAP-MRI to obtain both Tractography and Microstructural Contrasts from a Clinical DMRI Acquisition

Participants : Rutger H.j Fick, Mario Zuccheli [Dpt of Computer Science, University of Verona] , Gabriel Girard [SCIL Lab., Sherbrooke University] , Maxime Descoteaux [SCIL Lab., Sherbrooke University] , Gloria Menegaz [Dpt of Computer Science, University of Verona] , Rachid Deriche.

Diffusion MRI (dMRI) is used to characterize the directional- ity and microstructural properties of brain white matter (WM) by measuring the diffusivity of water molecules. In clinical practice the number of dMRI samples that can be obtained is limited, and one often uses short scanning protocols that ac- quire just 32 to 64 different gradient directions using a single gradient strength (b-value). Such 'single shell' scanning protocols restrict one to use methods that have assumptions on the radial decay of the dMRI signal over different b-values, which introduces estimation biases. In this work, we show, that by simply spreading the same number of samples over multiple b-values (i.e. multi-shell) we can accurately estimate both the WM directionality using 3D-SHORE and characterize the radially dependent diffusion microstructure measures using MAP-MRI. We validate our approach by undersampling both noisy synthetic and human brain data of the Human Connectome Project, proving this approach is well-suited for clinical applications.

This work has been submitted to ISBI'2015.

Laplacian-Regularized MAP-MRI Improving Axonal Caliber Estimation

Participants : Rutger H.j Fick, Demian Wassermann, Gonzalo Sanguinetti, Rachid Deriche.

In diffusion MRI, the accurate description of the entire diffusion signal from sparse measurements is essential to enable the recovery of microstructural information of the white matter. The recent Mean Apparent Propagator (MAP)-MRI basis is especially well suited for this task, but the basis fitting becomes unreliable in the presence of noise. As a solution we propose a fast and robust analytic Laplacian regularization for MAP-MRI. Using both synthetic diffusion data and human data from the Human Connectome Project we show that (1) MAP-MRI has more accurate microstructure recovery com- pared to classical techniques, (2) regularized MAP-MRI has lower signal fitting errors compared to the unregularized approach and a positivity constraint on the EAP and (3) that our regularization improves axon radius recovery on human data.

This work has been submitted to ISBI'2015.

A Unifying Framework for Spatial and Temporal Diffusion in Diffusion MRI

Participants : Rutger H.j Fick, Demian Wassermann, Marco Pizzolato, Rachid Deriche.

We propose a novel framework to simultaneously represent the diffusion-weighted MRI (dMRI) signal over diffusion times, gradient strengths and gradient directions. Current frameworks such as the 3D Simple Harmonic Oscillator Reconstruction and Estimation basis (3D- SHORE) only represent the signal over the spatial domain, leaving the temporal dependency as a fixed parameter. However, microstructure- focused techniques such as Axcaliber and ActiveAx provide evidence of the importance of sampling the dMRI space over diffusion time. Up to now there exists no generalized framework that simultaneously models the dependence of the dMRI signal in space and time. We use a functional basis to fit the 3D+t spatio-temporal dMRI signal, similarly to the 3D-SHORE basis in three dimensional 'q-space'. The lowest order term in this expansion contains an isotropic diffusion tensor that characterizes the Gaussian displacement distribution, multiplied by a negative exponential. We regularize the signal fitting by minimizing the norm of the analytic Laplacian of the basis. The continuous 3D+t signal representation can provide new insights on the anomalous nature of the dMRI signal in human tissues, i.e., when mean-squared molecular displacements varies slower than linearly with the diffusion time. From the fitting one can also estimate the axon radius distribution parameters along any direction using approaches similar to AxCaliber. We validate our technique on synthetic data generated using the theoretical model proposed by Callaghan et al. We show that our method is robust to noise and can accurately describe the restricted spatio-temporal signal decay originating from tissue models such as cylindrical pores. Moreover, we apply our method on real data from an ActiveAx acquisition. Overall our approach allows to represent the complete 3D+t dMRI signal which should prove helpful in understanding normal and pathologic nervous tissue.

This work has been submitted to IPMI'2015.

Fast and Robust EAP reconstruction using a Laplacian Regularized SHORE basis and its Impact on Microstructure Recovery

Participants : Rutger H.j Fick, Demian Wassermann, Emmanuel Caruyer [SBIA, University of Pennsylvania Medical School] , Rachid Deriche.

In diffusion MRI, the reconstructed Ensemble Average Propagator (EAP) from the diffusion signal provides detailed insights on the diffusion process and the underlying tissue microstructure. Recently, the 3D Simple Harmonic Oscillator based Reconstruction and Estimation (3D-SHORE) basis was proposed as a promising method to reconstruct the EAP. However, the fitting of the basis is sensitive to noise. To solve this we propose to use the Laplacian of the SHORE basis as a natural regularization functional. We provide the derivation of the Laplacian functional and compare its effect on EAP reconstruction with that of separated regularization of the radial and angular parts of the SHORE basis and imposing positive-definiteness in the estimation of the EAP. We validate our method on phantom data with known ground truth and on human data from the Human Connectome Project. We show that Laplacian regularization of the 3D-SHORE basis provides faster and more accurate estimation of the signal and EAP.

This work has been submitted to NeuroImage.